A391533 → Pₙ + Pₙ₊₁ → basis A013611

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, 0]coefficients of one sign Hadamard kernel

Recurrence

\(P_{n}(x) = \left(2 + 4\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 27 + 108\,x + 144\,x^{2} + 64\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	x¹⁷	P_n(1)
0	27	108	144	64															343
1	54	324	720	704	256														2058
2	108	864	2736	4288	3328	1024													12348
3	216	2160	8928	19520	23808	15360	4096												74088
4	432	5184	26496	74752	125696	125952	69632	16384											444528
5	864	12096	73728	255488	550400	754688	643072	311296	65536										2667168
6	1728	27648	195840	805888	2122752	3710976	4304896	3194880	1376256	262144									16003008
7	3456	62208	502272	2395136	7469056	15912960	23453696	23609344	15532032	6029312	1048576								96018048
8	6912	138240	1253376	6799360	24518656	61702144	110559232	141033472	125501440	74186752	26214400	4194304							576108288
9	13824	304128	3059712	18612224	76234752	221478912	467927040	724303872	815136768	650379264	349175808	113246208	16777216						3456649728
10	27648	663552	7335936	49463296	226918400	747896832	1821769728	3320315904	4527489024	4561305600	3299868672	1623195648	486539264	67108864					20739898368
11	55296	1437696	17326080	128270336	651689984	2403467264	6635126784	13927710720	22336241664	27232567296	24844959744	16445865984	7465861120	2080374784	268435456				124439390208
12	110592	3096576	40402944	325844992	1816461312	7413694464	22884122624	54395928576	100383326208	143810101248	158620188672	132271570944	80715186176	34024194048	8858370048	1073741824			746636341248
13	221184	6635520	93192192	813301760	4936302592	22093234176	75423023104	200328347648	418350366720	689153507328	892480782336	899023896576	690516656128	390909132800	153813516288	37580963840	4294967296		4479818047488
14	442368	14155776	212926464	1999372288	13125812224	63931678720	239218982912	702348787712	1638014124032	3051708481536	4541575593984	5367970922496	4977128898560	3543884890112	1871263563776	690415992832	158913789952	17179869184	26878908284928

Technical details

Node id: n-3eb516a297f686ad · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-f6df6c2d28e8f6fb

Observed on the stored rows ?

Degrees: 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 (affine: slope 1, intercept 3)
Palindromic rows: no
γ-positive (gamma_positive): no
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:    27,    108,     144,       64;
  1:    54,    324,     720,      704,      256;
  2:   108,    864,    2736,     4288,     3328,      1024;
  3:   216,   2160,    8928,    19520,    23808,     15360,      4096;
  4:   432,   5184,   26496,    74752,   125696,    125952,     69632,     16384;
  5:   864,  12096,   73728,   255488,   550400,    754688,    643072,    311296,     65536;
  6:  1728,  27648,  195840,   805888,  2122752,   3710976,   4304896,   3194880,   1376256,    262144;
  7:  3456,  62208,  502272,  2395136,  7469056,  15912960,  23453696,  23609344,  15532032,   6029312,   1048576;
  8:  6912, 138240, 1253376,  6799360, 24518656,  61702144, 110559232, 141033472, 125501440,  74186752,  26214400,   4194304;
  9: 13824, 304128, 3059712, 18612224, 76234752, 221478912, 467927040, 724303872, 815136768, 650379264, 349175808, 113246208, 16777216;

%F Formula

The row polynomials satisfy P(n,x) = (2+4*x)*P(n-1,x) for n >= 1, with P(0,x) = 27 + 108*x + 144*x^2 + 64*x^3.

%t Mathematica

P[0, x_] := 27 + 108*x + 144*x^2 + 64*x^3;
P[n_, x_] := P[n, x] = Expand[(2 + 4*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 27 + 108*x + 144*x^2 + 64*x^3; for(n = 1, N, my(i = n + 1); v[i] = (2 + 4*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from shift_x_plus_one of Riordan array ((1+x)^3,x).?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Riordan array ((1+x)^3,x).?
    
    Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
  - Geometric basis map (elementary) polytool ? step ›
    
    from Riordan array ((1+x)^3,x).?
    
    ↑ Row-hypergeometric rows (multiplier sequences) (above)
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.